Problem 38 - Precalculus: Functions and Graphs

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$y^{2}=\frac{14}{3}-x^{2}$$

Step-by-Step Solution

Answer

The graph is a circle.

Step 1: Identify General Form

The given equation is \( y^2 = \frac{14}{3} - x^2 \). Rearrange it to match the general form \( Ax^2 + By^2 = C \). Here, you can rewrite it as \( x^2 + y^2 = \frac{14}{3} \).

Step 2: Compare to Conic Section Forms

The equation \( x^2 + y^2 = C \) fits the general form of a circle which is \( x^2 + y^2 = r^2 \). Here, \( C = \frac{14}{3} \). Thus, this is a circle with radius \( \sqrt{\frac{14}{3}} \).

Step 3: State the Type of Conic Section

Since the equation can be expressed as \( x^2 + y^2 = \frac{14}{3} \), which follows the form of a circle, we identify this curve as a circle. The lack of x and y terms (like 2xy or individual x, y terms) confirms that the radius is the only important parameter here.

Key Concepts

Identifying Conic Sections

Conic sections are curves that can be formed by intersecting a plane with a double-napped cone. These sections include different types of curves: circles, ellipses, parabolas, and hyperbolas. To identify a conic section from an equation, you need to recognize its standard form.

Here's a simple breakdown:

• Circle: The equation is of the form \(x^2 + y^2 = r^2\).
• Ellipse: Looks like \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). If \(a = b\), it's a circle.
• Parabola: Equation form is \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\).
• Hyperbola: Has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Knowing these forms helps you quickly categorize the conic section. For example, if you see an equation like \(x^2 + y^2 = C\), you can immediately spot that it represents a circle.

Equations of Circles

The equation of a circle is a specific type of conic section equation. It is typically written in the form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center, and \(r\) is the radius. If the circle is centered at the origin (0, 0), the equation simplifies to \(x^2 + y^2 = r^2\).

To determine a circle from an equation, ensure there is no cross term \(xy\) or separate linear terms in \(x\) or \(y\). In the case of the example equation \(x^2 + y^2 = \frac{14}{3}\):

• This equation matches the form for a circle centered at the origin.
• The radius \(r\) of the circle can be calculated as \(\sqrt{\frac{14}{3}}\).

This equation's simplicity helps in quickly identifying it as a circle because it lacks any linear offset or complex terms.

Precalculus

Precalculus is a course that prepares students for calculus by covering various mathematical concepts, including algebra, trigonometry, and the study of functions and their graphs. A significant part of precalculus is understanding conic sections, which lay the foundational skills for calculus.

In precalculus:

• Students learn to identify and graph conic sections, such as circles, ellipses, parabolas, and hyperbolas.
• The course emphasizes manipulating and transforming equations to meet standard forms, aiding in graphing and analysis.
• Special attention is given to understanding the properties and applications of these curves.

Grasping these concepts is crucial as they frequently appear in calculus, engineering, physics, and other scientific fields that involve complex motion and design modeling. Developing skills in precalculus provides a robust mathematical toolkit for tackling these advanced studies.